Is there any GMP logarithm function?

I know you didn't ask how to implement it, but...

You can implement a rough one using the properties of logarithms: http://gnumbers.blogspot.com.au/2011/10/logarithm-of-large-number-it-is-not.html

And the internals of the GMP library: https://gmplib.org/manual/Integer-Internals.html

(Edit: Basically you just use the most significant "digit" of the GMP representation since the base of the representation is huge B^N is much larger than B^{N-1})

Here is my implementation for Rationals.

    double LogE(mpq_t m_op)
    {
        // log(a/b) = log(a) - log(b)
        // And if a is represented in base B as:
        // a = a_N B^N + a_{N-1} B^{N-1} + ... + a_0
        // => log(a) \approx log(a_N B^N)
        // = log(a_N) + N log(B)
        // where B is the base; ie: ULONG_MAX

        static double logB = log(ULONG_MAX);

        // Undefined logs (should probably return NAN in second case?)
        if (mpz_get_ui(mpq_numref(m_op)) == 0 || mpz_sgn(mpq_numref(m_op)) < 0)
            return -INFINITY;               

        // Log of numerator
        double lognum = log(mpq_numref(m_op)->_mp_d[abs(mpq_numref(m_op)->_mp_size) - 1]);
        lognum += (abs(mpq_numref(m_op)->_mp_size)-1) * logB;

        // Subtract log of denominator, if it exists
        if (abs(mpq_denref(m_op)->_mp_size) > 0)
        {
            lognum -= log(mpq_denref(m_op)->_mp_d[abs(mpq_denref(m_op)->_mp_size)-1]);
            lognum -= (abs(mpq_denref(m_op)->_mp_size)-1) * logB;
        }
        return lognum;
    }

(Much later edit) Coming back to this 5 years later, I just think it's cool that the core concept of log(a) = N log(B) + log(a_N) shows up even in native floating point implementations, here is the glibc one for ia64 And I used it again after encountering this question

No there is no such function in GMP. Only in MPFR.

The method below makes use of mpz_get_d_2exp and was obtained from the gmp R package. It can be found under the function biginteger_log in the file bigintegerR.cc (You first have to download the source (i.e. the tar file)). You can also see it here: biginteger_log.

// Adapted for general use from the original biginteger_log
// xi = di * 2 ^ ex  ==> log(xi) = log(di) + ex * log(2)

double biginteger_log_modified(mpz_t x) {
    signed long int ex;
    const double di = mpz_get_d_2exp(&ex, x);
    return log(di) + log(2) * (double) ex;
}

Of course, the above method could be modified to return the log with any base using the properties of logarithm (e.g. the change of base formula).

Here it is: https://github.com/linas/anant

Provides gnu mp real and complex logarithm, exp, sine, cosine, gamma, arctan, sqrt, polylogarithm Riemann and Hurwitz zeta, confluent hypergeometric, topologists sine, and more.